Propositional directed acyclic graph

A propositional directed acyclic graph (PDAG) is a data structure that is used to represent a Boolean function. A Boolean function can be represented as a rooted, directed acyclic graph of the following form:

Leaves labeled with \top (\bot) represent the constant Boolean function which always evaluates to 1 (0). A leaf labeled with a Boolean variable x is interpreted as the assignment x=1, i.e. it represents the Boolean function which evaluates to 1 if and only if x=1. The Boolean function represented by a \bigtriangleup-node is the one that evaluates to 1, if and only if the Boolean function of all its children evaluate to 1. Similarly, a \bigtriangledown-node represents the Boolean function that evaluates to 1, if and only if the Boolean function of at least one child evaluates to 1. Finally, a \Diamond-node represents the complemenatary Boolean function its child, i.e. the one that evaluates to 1, if and only if the Boolean function of its child evaluates to 0.

PDAG, BDD, and NNF

Every binary decision diagram (BDD) and every negation normal form (NNF) are also a PDAG with some particular properties. The following pictures represent the Boolean function f(x1, x2, x3) = -x1 * -x2 * -x3 + x1 * x2 + x2 * x3:

BDD for the function f
PDAG for the function f obtained from the BDD
PDAG for the function f

See also

References

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